Properties

Label 1617.2.a.x
Level $1617$
Weight $2$
Character orbit 1617.a
Self dual yes
Analytic conductor $12.912$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} - \beta_1) q^{5} - \beta_1 q^{6} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_{2} - \beta_1) q^{5} - \beta_1 q^{6} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{8} + q^{9} + (\beta_{3} - 2) q^{10} - q^{11} + ( - \beta_{2} - \beta_1 - 1) q^{12} - \beta_{3} q^{13} + ( - \beta_{2} + \beta_1) q^{15} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 3) q^{16} + (2 \beta_{2} + \beta_1) q^{17} + \beta_1 q^{18} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{19} + ( - \beta_{2} - \beta_1) q^{20} - \beta_1 q^{22} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{23} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{24} + ( - 2 \beta_{3} - 2 \beta_{2} - 1) q^{25} + ( - \beta_{2} + \beta_1) q^{26} - q^{27} + ( - \beta_{3} - \beta_1 + 2) q^{29} + ( - \beta_{3} + 2) q^{30} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{31} + (4 \beta_{2} + 5 \beta_1 + 6) q^{32} + q^{33} + (2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 5) q^{34} + (\beta_{2} + \beta_1 + 1) q^{36} + ( - \beta_{2} - \beta_1 - 1) q^{37} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{38} + \beta_{3} q^{39} + ( - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{40} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + ( - 2 \beta_{2} + 3 \beta_1 + 2) q^{43} + ( - \beta_{2} - \beta_1 - 1) q^{44} + (\beta_{2} - \beta_1) q^{45} + ( - 4 \beta_{2} + \beta_1 - 6) q^{46} + ( - \beta_{2} - \beta_1 - 3) q^{47} + ( - 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 3) q^{48} + ( - 2 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{50} + ( - 2 \beta_{2} - \beta_1) q^{51} + (\beta_{3} + 2) q^{52} + (4 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{53} - \beta_1 q^{54} + ( - \beta_{2} + \beta_1) q^{55} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{57} + ( - 2 \beta_{2} + 2 \beta_1 - 3) q^{58} + ( - 2 \beta_{3} + 2 \beta_1 - 5) q^{59} + (\beta_{2} + \beta_1) q^{60} + ( - \beta_{3} - 2 \beta_1) q^{61} + (\beta_{3} + 6 \beta_1 + 4) q^{62} + (5 \beta_{2} + 7 \beta_1 + 13) q^{64} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{65} + \beta_1 q^{66} + (2 \beta_{3} + \beta_{2} - 3 \beta_1 + 10) q^{67} + (3 \beta_{3} + 4 \beta_{2} + 7 \beta_1 + 12) q^{68} + (2 \beta_{3} + 2 \beta_1 - 1) q^{69} + (2 \beta_{3} + \beta_{2} - \beta_1 + 5) q^{71} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{72} + (\beta_{3} - 6 \beta_{2} - 4) q^{73} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 4) q^{74} + (2 \beta_{3} + 2 \beta_{2} + 1) q^{75} + (2 \beta_{2} + 3 \beta_1 + 4) q^{76} + (\beta_{2} - \beta_1) q^{78} + ( - 3 \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 2) q^{79} + ( - 2 \beta_{3} - 5 \beta_{2} + \beta_1 - 8) q^{80} + q^{81} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 4) q^{82} + ( - 4 \beta_{3} - 2 \beta_1 + 2) q^{83} + ( - \beta_{3} - 4 \beta_{2} + 2) q^{85} + ( - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 7) q^{86} + (\beta_{3} + \beta_1 - 2) q^{87} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{88} + ( - 6 \beta_{2} + 2 \beta_1 - 6) q^{89} + (\beta_{3} - 2) q^{90} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{92} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{93} + ( - \beta_{3} - 2 \beta_{2} - 5 \beta_1 - 4) q^{94} + ( - 5 \beta_{3} - 4 \beta_{2} + 4) q^{95} + ( - 4 \beta_{2} - 5 \beta_1 - 6) q^{96} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 11) q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 12 q^{8} + 4 q^{9} - 10 q^{10} - 4 q^{11} - 4 q^{12} + 2 q^{13} + 4 q^{15} + 12 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{22} + 4 q^{23} - 12 q^{24} + 4 q^{25} + 4 q^{26} - 4 q^{27} + 8 q^{29} + 10 q^{30} + 12 q^{31} + 26 q^{32} + 4 q^{33} + 16 q^{34} + 4 q^{36} - 4 q^{37} - 8 q^{38} - 2 q^{39} + 6 q^{40} - 2 q^{41} + 18 q^{43} - 4 q^{44} - 4 q^{45} - 14 q^{46} - 12 q^{47} - 12 q^{48} + 2 q^{50} + 2 q^{51} + 6 q^{52} - 12 q^{53} - 2 q^{54} + 4 q^{55} - 4 q^{58} - 12 q^{59} - 2 q^{61} + 26 q^{62} + 56 q^{64} - 4 q^{65} + 2 q^{66} + 28 q^{67} + 48 q^{68} - 4 q^{69} + 12 q^{71} + 12 q^{72} - 6 q^{73} - 16 q^{74} - 4 q^{75} + 18 q^{76} - 4 q^{78} + 2 q^{79} - 16 q^{80} + 4 q^{81} + 12 q^{82} + 12 q^{83} + 18 q^{85} + 36 q^{86} - 8 q^{87} - 12 q^{88} - 8 q^{89} - 10 q^{90} - 16 q^{92} - 12 q^{93} - 20 q^{94} + 34 q^{95} - 26 q^{96} + 44 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.51658
−0.552409
1.28734
2.78165
−1.51658 −1.00000 0.300014 2.33317 1.51658 0 2.57816 1.00000 −3.53844
1.2 −0.552409 −1.00000 −1.69484 −1.59002 0.552409 0 2.04107 1.00000 0.878345
1.3 1.28734 −1.00000 −0.342766 −3.91744 −1.28734 0 −3.01593 1.00000 −5.04306
1.4 2.78165 −1.00000 5.73760 −0.825711 −2.78165 0 10.3967 1.00000 −2.29684
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1617.2.a.x 4
3.b odd 2 1 4851.2.a.bu 4
7.b odd 2 1 1617.2.a.z 4
7.d odd 6 2 231.2.i.e 8
21.c even 2 1 4851.2.a.bt 4
21.g even 6 2 693.2.i.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.e 8 7.d odd 6 2
693.2.i.i 8 21.g even 6 2
1617.2.a.x 4 1.a even 1 1 trivial
1617.2.a.z 4 7.b odd 2 1
4851.2.a.bt 4 21.c even 2 1
4851.2.a.bu 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1617))\):

\( T_{2}^{4} - 2T_{2}^{3} - 4T_{2}^{2} + 4T_{2} + 3 \) Copy content Toggle raw display
\( T_{5}^{4} + 4T_{5}^{3} - 4T_{5}^{2} - 20T_{5} - 12 \) Copy content Toggle raw display
\( T_{13}^{4} - 2T_{13}^{3} - 8T_{13}^{2} + 16T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{4} + 2T_{17}^{3} - 40T_{17}^{2} - 124T_{17} + 15 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 4 T^{2} + 4 T + 3 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} - 4 T^{2} - 20 T - 12 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} - 8 T^{2} + 16 T - 4 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 40 T^{2} - 124 T + 15 \) Copy content Toggle raw display
$19$ \( T^{4} - 40 T^{2} + 122 T - 89 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} - 42 T^{2} + 76 T + 465 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} + 12 T^{2} + 14 T + 3 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} - 8 T^{2} + \cdots - 1396 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} - 10 T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{3} - 44 T^{2} + 64 T + 60 \) Copy content Toggle raw display
$43$ \( T^{4} - 18 T^{3} + 60 T^{2} + \cdots - 1385 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + 38 T^{2} + 40 T + 9 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} - 124 T^{2} + \cdots + 6252 \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} - 18 T^{2} + \cdots + 249 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} - 24 T^{2} - 32 T + 20 \) Copy content Toggle raw display
$67$ \( T^{4} - 28 T^{3} + 192 T^{2} + \cdots - 1460 \) Copy content Toggle raw display
$71$ \( T^{4} - 12 T^{3} + 2 T^{2} + 296 T - 699 \) Copy content Toggle raw display
$73$ \( T^{4} + 6 T^{3} - 272 T^{2} + \cdots + 17156 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} - 148 T^{2} + \cdots + 388 \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} - 96 T^{2} + \cdots + 2592 \) Copy content Toggle raw display
$89$ \( T^{4} + 8 T^{3} - 232 T^{2} + \cdots + 12048 \) Copy content Toggle raw display
$97$ \( T^{4} - 44 T^{3} + 682 T^{2} + \cdots + 8501 \) Copy content Toggle raw display
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